ON THE WEAK-COUPLING LIMIT FOR A FERMI GAS IN A RANDOM POTENTIAL

General conditions are derived for the essential self-adjointness of - DELTA + V, where V is a translation-invariant random potential, and fo r the existence of the perturbation expansion. A sequence of graphs is exhibited violating Dell'Antonio's bound for skeleton graphs. For a t ranslation-invariant and clustering Gaussian random potential V, and a translation-invariant and clustering initial state S of the Fermi gas , uncorrelated with the random potential, the weak coupling limit (Van Hove limit) yields increase of entropy, propagation of chaos, converg ence of the state for sufficiently small values of the parameter T to a gauge-invariant and quasi-free asymptotic state, and the semigroup d escribing the evolution of the two-point function. The asymptotic syst em is Bernoulli. Results are obtained not only for the average over th e random potential but also with probability one. If the random potent ial V' is absolutely continuous with respect to V, and if the state S' is given by a density matrix in the GNS representation for S, then th e weak coupling limit is the same as for V and S.